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Theorem vtoclr 4331
Description: Variable to class conversion of transitive relation. (Contributed by NM, 9-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypotheses
Ref Expression
vtoclr.1 Rel 𝑅
vtoclr.2 ((x𝑅y y𝑅z) → x𝑅z)
Assertion
Ref Expression
vtoclr ((A𝑅B B𝑅𝐶) → A𝑅𝐶)
Distinct variable groups:   x,y,A   y,B   x,z,𝐶,y   x,𝑅,y,z
Allowed substitution hints:   A(z)   B(x,z)

Proof of Theorem vtoclr
StepHypRef Expression
1 vtoclr.1 . . . . . 6 Rel 𝑅
21brrelexi 4327 . . . . 5 (A𝑅BA V)
31brrelex2i 4328 . . . . 5 (A𝑅BB V)
42, 3jca 290 . . . 4 (A𝑅B → (A V B V))
51brrelex2i 4328 . . . 4 (B𝑅𝐶𝐶 V)
6 breq1 3758 . . . . . . . 8 (x = A → (x𝑅yA𝑅y))
76anbi1d 438 . . . . . . 7 (x = A → ((x𝑅y y𝑅𝐶) ↔ (A𝑅y y𝑅𝐶)))
8 breq1 3758 . . . . . . 7 (x = A → (x𝑅𝐶A𝑅𝐶))
97, 8imbi12d 223 . . . . . 6 (x = A → (((x𝑅y y𝑅𝐶) → x𝑅𝐶) ↔ ((A𝑅y y𝑅𝐶) → A𝑅𝐶)))
109imbi2d 219 . . . . 5 (x = A → ((𝐶 V → ((x𝑅y y𝑅𝐶) → x𝑅𝐶)) ↔ (𝐶 V → ((A𝑅y y𝑅𝐶) → A𝑅𝐶))))
11 breq2 3759 . . . . . . . 8 (y = B → (A𝑅yA𝑅B))
12 breq1 3758 . . . . . . . 8 (y = B → (y𝑅𝐶B𝑅𝐶))
1311, 12anbi12d 442 . . . . . . 7 (y = B → ((A𝑅y y𝑅𝐶) ↔ (A𝑅B B𝑅𝐶)))
1413imbi1d 220 . . . . . 6 (y = B → (((A𝑅y y𝑅𝐶) → A𝑅𝐶) ↔ ((A𝑅B B𝑅𝐶) → A𝑅𝐶)))
1514imbi2d 219 . . . . 5 (y = B → ((𝐶 V → ((A𝑅y y𝑅𝐶) → A𝑅𝐶)) ↔ (𝐶 V → ((A𝑅B B𝑅𝐶) → A𝑅𝐶))))
16 breq2 3759 . . . . . . . 8 (z = 𝐶 → (y𝑅zy𝑅𝐶))
1716anbi2d 437 . . . . . . 7 (z = 𝐶 → ((x𝑅y y𝑅z) ↔ (x𝑅y y𝑅𝐶)))
18 breq2 3759 . . . . . . 7 (z = 𝐶 → (x𝑅zx𝑅𝐶))
1917, 18imbi12d 223 . . . . . 6 (z = 𝐶 → (((x𝑅y y𝑅z) → x𝑅z) ↔ ((x𝑅y y𝑅𝐶) → x𝑅𝐶)))
20 vtoclr.2 . . . . . 6 ((x𝑅y y𝑅z) → x𝑅z)
2119, 20vtoclg 2607 . . . . 5 (𝐶 V → ((x𝑅y y𝑅𝐶) → x𝑅𝐶))
2210, 15, 21vtocl2g 2611 . . . 4 ((A V B V) → (𝐶 V → ((A𝑅B B𝑅𝐶) → A𝑅𝐶)))
234, 5, 22syl2im 34 . . 3 (A𝑅B → (B𝑅𝐶 → ((A𝑅B B𝑅𝐶) → A𝑅𝐶)))
2423imp 115 . 2 ((A𝑅B B𝑅𝐶) → ((A𝑅B B𝑅𝐶) → A𝑅𝐶))
2524pm2.43i 43 1 ((A𝑅B B𝑅𝐶) → A𝑅𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551   class class class wbr 3755  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by:  domtr  6201
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